    Accelerated plane-wave destruction  Next: Accelerated PWD Up: Theory Previous: Theory

## Review of PWD

The local plane wave can be represented by the following differential equation (Claerbout, 1992): (1)

where is the local slope in continuous space, with dimension time/length. The wavefields observed at the two positions have a time delay which is proportional to their distance, . In the sampled system with space and time intervals and , we define the discrete space slope in the unit of , as . As is independent of the sampling interval, it can be directly used in irregular dataset (in this case, the unit of the slopes becomes space variant). The time delay between two adjacent positions is then the slope : (2)

With the transform applied along both time and space directions, the above equation becomes (3)

where is the unit time-shift operator, denotes the unit space-shift operator and is the transform of . The operator is the plane-wave destructor. Using Thiran's fractional delay filter (Thiran, 1971) to approximate the time-shift operator , where is the circular frequency, the plane-wave destructor can be expressed as (Fomel, 2002), (4)

where (5) is the order of the noncausal temporal filter and are functions of the local slope .

Equation 4 is a 2D filter. Applying the filter at an arbitrary point in the wavefield, the plane-wave destruction equation 3 becomes a nonlinear equation for the local slope : (6)

An iterative method, such as Newton's method, can be applied to find the slope. In practice, wavefields are polluted by noise and the plane wave assumption may not hold true where faults and conflicting boundaries exist. To obtain a stable slope estimation, an additional smoothing regularization process (Fomel, 2007a) is needed at each step. The total computational cost of slope estimation by plane-wave destruction becomes , where is the size of the data, is the size of the filter, is the number of linear iterations for regularization, and is the number of nonlinear iterations for solving equation 6. Typical values are , -50, and -10.    Accelerated plane-wave destruction  Next: Accelerated PWD Up: Theory Previous: Theory

2013-03-02